From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math,sci.electronics.misc,alt.sci.physics.acoustics
Subject: Piano tuning (was Re: Pythagorean comma)
Date: 30 Jan 1996 16:43:03 GMT
In article <4ejgp4$bns@geraldo.cc.utexas.edu>, Bruce Bostwick
wrote:
>seriously, though, piano tuners can detect differences in pitch of
>less than a hertz (yes, I do mean one cycle per second!) and the good
>ones can match pitches to within a hundred ppm or better.
Really this is not so much a good ear as good trigonometry, and I
subject students to it from time to time.
Suppose you have two sine waves (pure, stable tones) of equal amplitude
(loudness) but slightly different frequency (pitch). This means they
are changing the air pressure by a multiple of
sin( f1 t + c1) and
sin( f2 t + c2) respectively.
The chord that you hear when the tones are sounded together is the
result of the sum of these changes in air pressure (superposition principle).
Writing these functions in the form sin( a + b ) and sin( a - b ) we
expand using the angle-addition formula and add to get 2 sin(a) cos(b),
that is,
[ 2 cos( (f1-f2)/2 * t + (c1-c2)/2 ) ] *
sin( (f1+f2)/2 * t + (c1+c2)/2 )
This may be viewed -- err, heard -- as a pure tone whose frequency
(f1+f2)/2 is midway between the other two (which is to say, it sounds
pretty much like either one of them separately), but which is not stable.
Indeed, its magnitude is slowly changing -- from being twice as loud
as either component, as you might expect, to nearly silent. The magnitude
goes through a complete cycle with frequency (f1-f2)/2. In particular,
for two tones whose frequency differs by 1 Hz, this results in a
sound whose amplitude drops to zero twice every 2-second period. In other
words, there will be a distinct warbling sound, as if the tone were
coming from a horn and the player were applying a mute off and on, once
per second. The sound is referred to as beats, and the frequency
(f1-f2) is the beat frequency.
A piano tuner thus need not be able to decide if each of the two or
three strings supposedly playing one note really produce the same
frequency. S/he need only play the two notes together, then sharpen or
flatten one so that beats are eliminated. No particularly good ear is
needed, although without training one will just as often flatten what
one should sharpen, since the beat-frequency method is only sensitive
to the _absolute value_ of f1-f2.
This same idea can be used to tune strings playing two different notes.
As has been noted earlier in this thread, an "ideal" fifth is a chord
using two frequencies in a 3:2 ratio. This is almost the change in
frequency between two notes which are 7 half-steps apart on the piano,
say between an A and an E. Unique Factorization being what it is,
however, this is not possible globally, and that interval on a piano
is usually tuned so that the ratio of frequencies is 2^(7/12):1, which
is about 0.12% too small.
The A near the middle of the keyboard is supposed to ring at 440 Hz,
and thus the E above it at 659.26 Hz. One may tune that E first to
660 Hz by first eliminating beats relative to the A, then flattening
the note (by loosening the string) until beats are heard at a rate of
about one every 1-1/3 second.
One can actually proceed through the whole 12-note octave with just the
single tuning fork in this way. The process is known as the Circle of Fifths.
It works because [7] is a generator of Z/12Z.
In practice of course things never go this smoothly. One must contend with
a number of issues, among them:
(1) A 1% difference in frequency produces a noticeable beat when the
tones are near the center of the keyboard. But when f1 = 50 Hz or 3500 Hz,
this becomes rather hard to pick up, and a nuisance to control.
(2) A tone on a real instrument is not a single frequency but the sum of
many, with different amplitudes. If this sounds like Fourier analysis, it
is, except that the frequencies need not all be integer multiples of the
lowest (i.e., "harmonics"). It's actually in the harmonics that one listens
for beats when tuning fifths, and when tuning low notes (see (1)).
But when the higher frequencies are not multiples of the fundamental, one
finds that two notes may be "in tune" with each other's fundamental
frequency but not with the higher ones.
(3) The trigonometry used above assumed the two waves were of equal magnitude.
This is not critical in practice, but it is worth noting that the
trigonometry is much murkier in general. For example,
sin( f1 x ) + 2 sin( f2 x )
does not factor as A* sin( f3 x ) * cos( f4 x), say. Indeed, you might
compute the first few zeros of the sum (try, say, f1= Pi and f2 = 0.97 Pi)
and you'll notice that they are not equally separated. You do hear something
like beat frequencies, but the frequency of the underlying tone also
changes in time. (Now it sounds more as if the horn player were swinging
on a swing and you had the Doppler effect to contend with too.)
At my home page there are a few other odds and ends as well as some
citations regarding some mathematical features of sound and music.
A direct URL is http://www.math.niu.edu/~rusin/papers/uses-math/music.
dave
==============================================================================
From: Bruce Bostwick
Newsgroups: sci.math,sci.electronics.misc,alt.sci.physics.acoustics
Subject: Re: Piano tuning (was Re: Pythagorean comma)
Date: 30 Jan 1996 22:13:18 GMT
rusin@washington.math.niu.edu (Dave Rusin) wrote a right scholarly reply
which was far too long for my article editor to quote :-( ...
Yes, the beat frequency is the absolute value of the pitch difference.
You do have to be very careful to remember which string is the reference
and which one is being tuned, and it is *easy* to go the wrong way
(broken a couple of bass strings that way!)
In the middle of the keyboard, the fifths *do* come out about a beat and
a half flat, and the strings have more than enough harmonics to make it
audible. It is hard to hear this ratio at the top or the bottom, but
you usually tune those by octaves anyway. In practice, it's a lot
easier to set the temperament in a two-octave scale by the circle of
fifths than it is to do anything else. You can do it by thirds, etc.
but it's a lot harder.
There is an *excellent* book by J. Cree Fischer on the subject, if
anyone is interested in further reading.
http://ccwf.cc.utexas.edu/~lihan/
==============================================================================
Date: Thu, 1 Feb 1996 10:27:52 -0600 (CST)
From: Bob Dillon
To: Dave Rusin
Subject: Piano tuning (was Re: Pythagorean comma)
Just a little side note in which you may be interested. In many small
twin engine airplanes, autosynchronizers are not installed, and
synchronization of the props is done by ear just as is your description
of piano tuning.
Bob
==============================================================================
Date: Thu, 1 Feb 96 10:32:37 CST
From: rusin (Dave Rusin)
To: bdillon@admin.aurora.edu
Subject: Re: Piano tuning (was Re: Pythagorean comma)
Cool! It's also true that this technique is the basis of FM (frequency
modulation) radio transmission: broadcast a wave of high frequency and
constant amplitude, but change the frequency (slowly); the radio receiver
buzzes at the nominal frequency (fixed) and then the trig identity produces
a "beat" frequency which flutters at the rate the two frequencies differ.
With a 100MHz nominal frequency, the frequencies of sound (say 100Hz -
10kHz) can easily be overlaid as a beat frequency.
dave
==============================================================================
Date: Fri, 22 May 1998 12:31:33 -0700
From: Fabio Rivera
To: rusin@math.niu.edu
Subject: Piano Tuning.
Dear Dr. Ruskin:
I am an applied math student at San Francisco State University in
California.
I am doing a project on the effect of octave stretching on equal
temperament tuning. In all the books of tuning theory I've come across,
perfectly tuned octaves are assumed; however, it seems that no tuners
actually tune pianos that way. My hope is to construct a model that
will account for this stretching.
My belief is that this stretching is only partially the result of
inharmonicity (the effect of string width) and that stretched octaves
provide a further correction to tuning. Most string players stretch
their notes at higher octaves, for example.
Have you come across any research on this topic?
I am: Tony Gualtieri
tonyg@sfsu.edu
==============================================================================
From: Dave Rusin
Date: Mon, 15 Jun 1998 01:11:36 -0500 (CDT)
To: bobafett@sfsu.edu
Subject: Re: Piano Tuning.
You sent me mail a few weeks ago on piano tuning. You asked in particular
about octave stretching.
It seems to me there are two issues here, physical and perceptual. On
the physical side we have the problem that real strings --
particularly metal ones -- don't have a well-defined pitch. If you
watch amplitudes of vibrations on an oscilloscope, you don't really
see a stable pattern; rather, there's a pattern, but it's changing
over time. Mathematically, this amounts to adding a sine wave with a
frequency which is not quite a mutliple of the fundamental frequency.
In practice this means that some strings cannot be tuned to other
strings or to a fixed ideal pitch.
I'm less clear on the perceptual end of things. The human auditory
system is pretty responsive over a wide range of frequencies (much more so
than the retina, for example) but not uniformly so. I'm under the impression
that to get a suitable response at the extreme ends one has to
magnify the sensation, stimulating the inner ear at frequencies a
little beyond the desired one in some sort of brain-fake. I know that
the people who design recording equipment (mikes, speakers, CDs)
understand all these perceptual issues -- I saw it in a FAQ once! -- so
I can only imagine musicians would have picked this up, too.
If you get better information I'd like to know it; I can add it to the site.
dave